On the Weight Distribution of Weights Less than $2w_{\min }$ in Polar Codes

Abstract

The number of low-weight codewords is critical to the performance of error-correcting codes. In 1970, Kasami and Tokura characterized the codewords of Reed-Muller (RM) codes whose weights are less than $2w_{\min }$, where $w_{\min }$ represents the minimum weight. In this paper, we extend their results to decreasing polar codes. We present the closed-form expressions for the number of codewords in decreasing polar codes with weights less than $2w_{\min }$. Moreover, the proposed enumeration algorithm runs in polynomial time with respect to the code length.